Feasibility of a team-based FPS AI?
To get a better understanding, let's compare a couple of games DOTA2 and Overwatch. At first, both seen similar in terms of multiplayer and strategy, but there is a significant difference.

You have mentioned it in the question. reaction time

Even though both the games require strategy and understanding of the surroundings and team play, FPS games are heavily dependent on reaction time and accuracy. As much as you try, it's next to impossible to reach the reaction times and accuracy of computer agent.

In games such as DOTA2, the game is more dependent towards strategy and how the characters interact with each other (Strengths and Weakness).

Finally, it doesn't make sense to train an AI based agent to play FPS games, because the optimal solution is already present and there is no way that AI can reach those levels. It's called AIM BOT. As good as you can get at strategy and planning, team work and reaction times, you can almost never beat it. It's next to 100% accurate and the opponent is basically dead as soon as he's in sight.